Step 1: Draw the ammonium ion.
$\text{NH}_4^+$ is tetrahedral, with the central nitrogen bonded to four hydrogen atoms.
Step 2: Look at the bond polarity.
Nitrogen is highly electronegative and the ion carries a positive charge, so every N-H bond is strongly polarised and each H gains a sizeable $\delta+$.
Step 3: Recall how a cation is solvated.
In water, the partially positive hydrogens attract the lone pairs on the oxygen of nearby water molecules, forming hydrogen bonds.
Step 4: Count the available donor hydrogens.
All four N-H hydrogens are equivalent and each one can donate to a separate water molecule.
Step 5: Picture the hydration shell.
Four water molecules cluster around the ion, one anchored at each of the four hydrogens.
Step 6: Tally the count.
That gives 4 hydrogen atoms engaged in hydrogen bonding during solvation.
Step 7: Select the answer.
The number is 4, which is option (2).
\[ \boxed{4\ \text{H atoms of NH}_4^+\ \text{H-bond to water}} \]